If G_1 and G_2 are two geometric means betwee | vedclass

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(C) Let the two numbers be $p$ and $q$.The two geometric means $G_1$ and $G_2$ between $p$ and $q$ are given by $G_1 = p(q/p)^{1/3} = p^{2/3}q^{1/3}$

Vedclass · Accepted Answer

$2A$

Vedclass · Answer

(C) Let the two numbers be $p$ and $q$.The two geometric means $G_1$ and $G_2$ between $p$ and $q$ are given by $G_1 = p(q/p)^{1/3} = p^{2/3}q^{1/3}$ and $G_2 = p(q/p)^{2/3} = p^{1/3}q^{2/3}$.Now,calculate the expression $\frac{G_1^2}{G_2} + \frac{G_2^2}{G_1}$:$\frac{G_1^2}{G_2} = \frac{(p^{2/3}q^{1/3})^2}{p^{1/3}q^{2/3}} = \frac{p^{4/3}q^{2/3}}{p^{1/3}q^{2/3}} = p^{4/3-1/3} = p$.Similarly,$\frac{G_2^2}{G_1} = \frac{(p^{1/3}q^{2/3})^2}{p^{2/3}q^{1/3}} = \frac{p^{2/3}q^{4/3}}{p^{2/3}q^{1/3}} = q^{4/3-1/3} = q$.Thus,$\frac{G_1^2}{G_2} + \frac{G_2^2}{G_1} = p + q$.Since the arithmetic mean $A = \frac{p+q}{2}$,we have $p+q = 2A$.Therefore,the value is $2A$.

If $G_1$ and $G_2$ are two geometric means between two numbers and $A$ is the arithmetic mean between them,then find the value of $\frac{G_1^2}{G_2} + \frac{G_2^2}{G_1}$.

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